I had a question on a math test which said that you should find an approximation for [tex] e^x [/tex] which is very good for [tex] x \approx 0 [/tex]. First I declared the function [tex] f(x) = e^x [/tex]. We have the intresting thing that [tex] f(x) = f'(x) = f''(x) = f'''(x) \ldots \ \forall x [/tex]. And because of this we have [tex] f(0) = f'(0) = f''(0) = f'''(0) \ldots [/tex]. Next I defined the function [tex] g(x) = ax^3+bx^2+cx+d[/tex] and if we want [tex] g(x) \approx f(x) [/tex] for [tex] x \approx 0 [/tex]. Using this it leads to that [tex] d=1 [/tex], [tex] c=1 [/tex], [tex] b = 1/2 [/tex], [tex] a=1/6 [/tex]. So [tex] g(x) = x^3/6 + x^2/2 + x + 1 [/tex]. This formula is good for [tex] x \approx 0 [/tex]. So I tried with [tex] h(x) = ax^4+bx^3+cx^2+dx+e [/tex] which leads to [tex] h(x) = x^4/24 + x^3/6 + x^2/2+x+1 [/tex] which is better. I found a pattern, we have for an [tex] j [/tex] degree polynom the formulas [tex] \sum_{u=0}^{j} \frac{ x^u}{u!} [/tex]. But graphing [tex] e^x [/tex] and a polynom of [tex] j [/tex] degree we get better and better result when [tex] j \to \infty [/tex]. So on my test I wrote done that a good approximation for [tex] e^x [/tex] for [tex] x \approx 0 [/tex] would be [tex] \sum_{u=0}^{\infty} \frac{x^u}{u!} [/tex]. I've just started calculus and that stuff, so I don't know if this is the answer my teacher wanted. I only know of factioral and sums because I got to study discreet math instead of psychology. Using the same technique as I used for finding an approximation for [tex] e^x [/tex] I gave formulas for [tex] \cos{x} [/tex] and [tex] \sin{x} [/tex]. The formulas are [tex] \sin{x} \approx \sum_{u=0}^{\infty} \frac{ \left( - 1 \right)^u \cdot x^{2u+1}}{ \left( 2u+1 \right)!} [/tex] and [tex] \cos{x} \approx \frac{\left(-1 \right)^u \cdot x^{2u}}{ \left( 2u \right)!} [/tex]. Are these formulas used for anything? And most important, are they correct - when I graph them they seem to be correct.(adsbygoogle = window.adsbygoogle || []).push({});

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# Approximations for e^x, sin and cos

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